State transition table

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In automata theory and sequential logic, a state transition table is a table showing what state (or states in the case of a nondeterministic finite automaton) the state machine will move to, based on the current state and other inputs. A state table is essentially a truth table in which some of the inputs are the current state, and the outputs include the next state, along with other outputs.

A state table is one of many ways to specify a state machine, other ways being a state diagram, and a characteristic equation.

1 Common forms
- 1.1 One-dimensional state tables
- 1.2 Two-dimensional state tables
2 Example
3 Transformations from/to state diagram
4 References

[edit] Common forms

[edit] One-dimensional state tables

Also called characteristic tables, single-dimension state tables are much more like truth tables than the two-dimensional versions. Inputs are usually placed on the left, and separated from the outputs, which are on the right. The outputs will represent the next state of the machine. Here's a simple example of a state machine with two states, and two combinatorial inputs:

A	B	Current State	Next State	Output
0	0	S₁	S₂	1
0	0	S₂	S₁	0
0	1	S₁	S₂	0
0	1	S₂	S₂	1
1	0	S₁	S₁	1
1	0	S₂	S₁	1
1	1	S₁	S₁	1
1	1	S₂	S₂	0

S₁ and S₂ would most likely represent the single bits 0 and 1, since a single bit can only have two states.

[edit] Two-dimensional state tables

State transition tables are typically two-dimensional tables. There are two common forms for arranging them.

The vertical (or horizontal) dimension indicates current states, the horizontal (or vertical) dimension indicates events, and the cells (row/column intersections) in the table contain the next state if an event happens (and possibly the action linked to this state transition).

**State Transition Table**
Events State	E₁	E₂	...	E_n
S₁	-	A_y/S_j	...	-
S₂	-	-	...	A_x/S_i
...	...	...	...	...
S_m	A_z/S_k	-	...	-

(S: state, E: event, A: action, -: illegal transition)

The vertical (or horizontal) dimension indicates current states, the horizontal (or vertical) dimension indicates next states, and the row/column intersections contain the event which will lead to a particular next state.

**State Transition Table**
next current	S₁	S₂	...	S_m
S₁	A_y/E_j	-	...	-
S₂	-	-	...	A_x/E_i
...	...	...	...	...
S_m	-	A_z/E_k	...	-

(S: state, E: event, A: action, -: impossible transition)

[edit] Example

An example of a state transition table for a machine M together with the corresponding state diagram is given below.

**State Transition Table**
Input State	1	0
S₁	S₁	S₂
S₂	S₂	S₁

State Diagram

All the possible inputs to the machine are enumerated across the columns of the table. All the possible states are enumerated across the rows. From the state transition table given above, it is easy to see that if the machine is in S₁ (the first row), and the next input is character 1, the machine will stay in S₁. If a character 0 arrives, the machine will transition to S₂ as can be seen from the second column. In the diagram this is denoted by the arrow from S₁ to S₂ labeled with a 0.

For a nondeterministic finite automaton (NFA), a new input may cause the machine to be in more than one state, hence its non-determinism. This is denoted in a state transition table by a pair of curly braces { } with the set of all target states between them. An example is given below.

**State Transition Table for an NFA**
Input State	1	0	ε
S₁	S₁	{ S₂, S₃ }	Φ
S₂	S₂	S₁	Φ
S₃	S₂	S₁	S₁

Here, a nondeterministic machine in the state S₁ reading an input of 0 will cause it to be in two states at the same time, the states S₂ and S₃. The last column defines the legal transition of states of the special character, ε. This special character allows the NFA to move to a different state when given no input. In state S₃, the NFA may move to S₁ without consuming an input character. The two cases above make the finite automaton described non-deterministic.

[edit] Transformations from/to state diagram

It is possible to draw a state diagram from the table. A sequence of easy to follow steps is given below:

Draw the circles to represent the states given.
For each of the states, scan across the corresponding row and draw an arrow to the destination state(s). There can be multiple arrows for an input character if the automaton is an NFA.
Designate a state as the start state. The start state is given in the formal definition of the automaton.
Designate one or more states as accept state. This is also given in the formal definition.